Rather than consider this quadratic scalar Lagrangian directly, we first add a convenient total derivative (which we determined through trial-and-error). Generally, adding a total derivative to a Lagrangian reorganizes how information is stored in the fields, but ultimately does not change the physics described by the Lagrangian; a classic example of this is the Gibbons-Hawking-York total derivative, which is used to make the Einstein-Hilbert Lagrangian of 4D gravity into a Lagrangian which only depends on fields and their first derivatives [1]}. The total derivative we add to \(\mathcal {L}_{\text{5D},rr}\)  (in addition to the total derivative \(\Delta \mathcal {L}\)  defined in Eq. (REF ), which is already folded into \(\mathcal {L}_{\text{5D},rr}\) ) is
\(\overline{\Delta } \mathcal {L}_{rr} &= \dfrac{1}{r_{c}^{2}} \partial _{\varphi } \bigg \lbrace  \hat{r}\,\hat{r}^{\,\prime } - \dfrac{3\, e^{2A}}{(\phi _{0}^{\prime })^{2}} \hat{z}\,\hat{r}^{\,\prime } + \dfrac{12}{(\phi _{0}^{\prime })^{4}} \Big [ A^{\prime } (\phi _{0}^{\prime })^{2} + V^{\prime } r_{c}^{2}\Big ] (\hat{r}^{\,\prime })^{2} \bigg \rbrace \) 
